Universal conductance at quantum-critical points

At PTB, the electrical unit of resistance, the ohm, is realized with high precision by means of the quantum Hall effect. An electric current is driven through the interface of a sample made of semiconductor layers that, at low temperatures, is exposed to a strong magnetic field. The ratio of the current flowing through the samples to the Hall voltage measured across the current direction defines the quantized Hall resistance. This quantity is determined only by the ratio of PlanckÕs constant h and the elementary charge e, as well as by an integral number. Presently, the conductance detected parallel to the current flow is of no importance for metrological applications, but it is essential for the understanding of the quantum Hall effect, as well as for the general theory of quantum phase transitions.

Up to now, ¹/₂ e²/h has been predicted for the maxima of the longitudinal conductivity that occur only at the steps of quantized Hall plateaus, which was also confirmed by earlier experiments. However, new quantum-mechanical model calculations have furnished a universal value of 0.6 e²/h for both the two-point conductance and the longitudinal conductivity, and this independent of the microscopic parameters selected.

The scale-invariant spatial structure of the electronic wave function at special energies Ð the so-called quantum-critical points Ð could be determined as the cause of this unexpected discrepancy. Although their multifractal properties have been known for some time already, it is only now that a quantitative relation between the experimentally accessible conductance and the theoretically calculated fractal dimensions could be established. This new finding has also been confirmed for the critical conductance at the metal-insulator transition of two-dimensional disordered systems with spin-orbit interaction. These are presently being investigated more intensively in relation with spintronic applications. The link between critical conductance and fractal dimensions appears to be a universal property of quantum phase transitions.